Towards the equation of state of quark-gluon matter from lattice QCD with Wilson twisted mass fermions

This contribution gives an overview of a long-term study of quark-gluon matter under extreme conditions employing Euclidean lattice discretized Quantum Chromodynamics (QCD) with Wilson twisted mass fermions and improved gauge actions. The analysis relies on methods and results of the European Twisted Mass Collaboration at zero temperature. We will start for two mass-degenerate light flavour degrees of freedom with a thorough investigation of the three-dimensional phase diagram of the bare coupling and the two mass-related parameters. Then for the same case we are going to describe the search for the pseudo-critical behaviour at various quark mass values. We shall discuss the behaviour of the (pseudo-)critical temperature Tc as a function of the (charged) pion mass mπ and the difficulty to extract the chiral limit in order to establish or reject the conjectured O(4)-universality. For the two-flavour case we shall further present very recent results for the thermodynamic Equation of State at non-zero temperature. Finally we will provide a view on an ongoing first exploration of QCD thermodynamics under the inclusion of virtual strange and charm quarks at their realistic mass values.